I am working in the US for a professor as a contracted research staff, and my contract will soon end. The professor holds daily meetings with the bosses of several companies, over a project we collaborate on. These companies have several public job listings that I consider myself a good fit for. Unfortunately, the scope of my job doesn't allow me much interaction with the staff from these companies.
However, I can see on LinkedIn that at least 500 people are applying to each of these positions every week. I am worried that if I file my application the conventional way, then it will get lost in the sea of HR paperwork, or even not get past the automatic filter. These companies also have a preference for PhDs, which I do not have at the moment.
Is it appropriate for me to persuade the professor to speak to the bosses directly about these jobs on my behalf? Are there any potential downsides?